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Lecture Notes on Nonstandard Analysis Ucla Summer School in Logic

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Lecture Notes on Nonstandard Analysis Ucla Summer School in Logic LECTURE NOTES ON NONSTANDARD ANALYSIS UCLA SUMMER SCHOOL IN LOGIC ISAAC GOLDBRING Contents 1. The hyperreals 3 1.1. Basic facts about the ordered real field 3 1.2. The nonstandard extension 4 1.3. Arithmetic in the hyperreals 5 ∗ 1.4. The structure of N 7 1.5. More practice with transfer 8 1.6. Problems 9 2. Logical formalisms for nonstandard extensions 10 2.1. Approach 1: The compactness theorem 11 2.2. Approach 2: The ultrapower construction 12 2.3. Problems 16 3. Sequences and series 17 3.1. First results about sequences 17 3.2. Cluster points 19 3.3. Series 21 3.4. Problems 22 4. Continuity 23 4.1. First results about continuity 23 4.2. Uniform continuity 25 4.3. Sequences of functions 27 4.4. Problems 30 5. Differentiation 33 5.1. The derivative 33 5.2. Continuous differentiability 35 5.3. Problems 36 6. Riemann Integration 38 6.1. Hyperfinite Riemann sums and integrability 38 6.2. The Peano Existence Theorem 41 6.3. Problems 43 7. Weekend Problem Set #1 44 8. Many-sorted and Higher-Type Structures 47 8.1. Many-sorted structures 47 Date: November 10, 2014. 1 2 ISAAC GOLDBRING 8.2. Higher-type sorts 48 8.3. Saturation 51 8.4. Useful nonstandard principles 53 8.5. Recap: the nonstandard setting 54 8.6. Problems 54 9. Metric Space Topology 55 9.1. Open and closed sets, compactness, completeness 55 9.2. More about continuity 63 9.3. Compact maps 64 9.4. Problems 65 10. Banach Spaces 67 10.1. Normed spaces 67 10.2. Bounded linear maps 68 10.3. Finite-dimensional spaces and compact linear maps 69 10.4. Problems 71 11. Hilbert Spaces 73 11.1. Inner product spaces 73 11.2. Orthonormal bases and `2 75 11.3. Orthogonal projections 79 11.4. Hyperfinite-dimensional subspaces 82 11.5. Problems 83 12. Weekend Problem Set #2 85 13. The Spectral Theorem for compact hermitian operators 88 13.1. Problems 93 14. The Bernstein-Robinson Theorem 94 15. Measure Theory 101 15.1. General measure theory 101 15.2. Loeb measure 102 15.3. Product measure 103 15.4. Integration 104 15.5. Conditional expectation 104 15.6. Problems 105 16. Szemer´ediRegularity Lemma 106 16.1. Problems 108 References 110 Nonstandard analysis was invented by Abraham Robinson in the 1960s as a way to rescue the na¨ıve use of infinitesimal and infinite elements favored by mathematicians such as Leibniz and Euler before the advent of the rigorous methods introduced by Cauchy and Weierstrauss. Indeed, Robinson realized that the compactness theorem of first-order logic could be used to provide fields that \logically behaved" like the ordered real field while containing \ideal" elements such as infinitesimal and infinite elements. LECTURE NOTES ON NONSTANDARD ANALYSIS 3 Since its origins, nonstandard analysis has become a powerful mathemat- ical tool, not only for yielding easier definitions for standard concepts and providing slick proofs of well-known mathematical theorems, but for also providing mathematicians with amazing new tools to prove theorems, e.g. hyperfinite approximation. In addition, by providing useful mathematical heuristics a precise language to be discussed, many mathematical ideas have been elucidated greatly. In these notes, we try and cover a wide spectrum of applications of non- standard methods. In the first part of these notes, we explain what a non- standard extension is and we use it to reprove some basic facts from calculus. We then broaden our nonstandard framework to handle more sophisticated mathematical situations and begin studying metric space topology. We then enter functional analysis by discussing Banach and Hilbert spaces. Here we prove our first serious theorems: the Spectral Theorem for compact hermit- ian operators and the Bernstein-Robinson Theorem on invariant subspaces; this latter theorem was the first major theorem whose first proof was non- standard. We then end by briefly discussing Loeb measure and using it to give a slick proof of an important combinatorial result, the Szemer´edi Regularity Lemma. Due to time limitations, there are many beautiful subjects I had to skip. In particular, I had to omit the nonstandard hull construction (although this is briefly introduced in the second weekend problem set) as well as applica- tions of nonstandard analysis to Lie theory (e.g. Hilbert's fifth problem), geometric group theory (e.g. asymptotic cones), and commutative algebra (e.g. bounds in the theory of polynomial rings). We have borrowed much of our presentation from two main sources: Gold- blatt's fantastic book [2] and Davis' concise [1]. Occasionally, I have bor- rowed some ideas from Henson's [3]. The material on Szemer´edi'sRegularity Lemma and the Furstenberg Correspondence come from Terence Tao's blog. I would like to thank Bruno De Mendonca Braga and Jonathan Wolf for pointing out errors in an earlier version of these notes. 1. The hyperreals 1.1. Basic facts about the ordered real field. The ordered field of real numbers is the structure (R; +; ·; 0; 1; <). We recall some basic properties: >0 • (Q is dense in R) for every r 2 R and every 2 R , there is q 2 Q such that jr − qj < ; • (Triangle Inequality) for every x; y 2 R, we have jx + yj ≤ jxj + jyj; >0 • (Archimedean Property) for every x; y 2 R , there is n 2 N such that nx > y. Perhaps the most important property of the ordered real field is Definition 1.1 (Completeness Property). If A ⊆ R is nonempty and bounded above, then there is a b 2 R such that: • for all a 2 A, we have a ≤ b (b is an upper bound for A); 4 ISAAC GOLDBRING • if a ≤ c for all a 2 A, then b ≤ c (b is the least upper bound for A). Such b is easily seen to be unique and is called the least upper bound of A, or the supremum of A, and is denote sup(A). Exercise 1.2. Show that if A is nonempty and bounded below, then A has a greatest lower bound. The greatest lower bound is also called the infimum of A and is denoted inf(A). 1.2. The nonstandard extension. In order to start \doing" nonstandard analysis as quickly as possible, we will postpone a formal construction of the nonstandard universe. Instead, we will pose some postulates that a nonstan- dard universe should possess, assume the existence of such a nonstandard universe, and then begin reasoning in this nonstandard universe. Of course, after we have seen the merits of some nonstandard reasoning, we will return and give a couple of rigorous constructions of nonstandard universes. ∗ We will work in a nonstandard universe R that has the following prop- erties: ∗ (NS1) (R; +; ·; 0; 1; <) is an ordered subfield of (R ; +; ·; 0; 1; <). ∗ ∗ (NS2) R has a positive infinitesimal element, that is, there is 2 R such >0 that > 0 but < r for every r 2 R . n (NS3) For every n 2 N and every function f : R ! R, there is a \natural ∗ n ∗ extension" f :(R ) ! R . The natural extensions of the field 2 ∗ operations +; · : R ! R coincide with the field operations in R . n ∗ ∗ n Similarly, for every A ⊆ R , there is a subset A ⊆ (R ) such that ∗ n A \ R = A. ∗ (NS4) R , equipped with the above assignment of extensions of functions and subsets, \behaves logically" like R. The last item in the above list is, of course, extremely vague and imprecise. We will need to discuss some logic in order to carefully explain what we mean by this. Roughly speaking, any statement that is expressible in first-order logic and mentioning only standard numbers is true in R if and only if it ∗ is true in R . This is often referred to as the Transfer Principle, although, ∗ logically speaking, we are just requiring that R , in a suitable first-order language, be an elementary extension of R. We will explain this in more detail later in these notes. That being said, until we rigorously explain the logical formalism of non- standard analysis, we should caution the reader that typical transferrable statements involve quantifiers over numbers and not sets of numbers. For example, the completeness property for R says that \for all sets of num- bers A that are nonempty and bounded above, sup(A) exists." This is an example of a statement that is not transferrable; see Exercise 1.8 below. ∗ Definition 1.3. R is called the ordered field of hyperreals. n Remark. If f : A ! R is a function, where A ⊆ R , we would like to also ∗ ∗ consider its nonstandard extension f : A ! R . We will take care of this matter shortly. LECTURE NOTES ON NONSTANDARD ANALYSIS 5 1.3. Arithmetic in the hyperreals. First, let's discuss some immediate ∗ consequences of the above postulates. Since R is an ordered field, we can start performing the field operations to our positive infinitesimal . For example, has an additive inverse −, which is then a negative infinitesimal. Also, we can consider π · ; it is reasonably easy to see that π · is also a positive infinitesimal. (This will also follow from a more general principle that we will shortly see.) −1 >0 Since 6= 0, it has a multiplicative inverse . For a given r 2 R , 1 −1 since < r , we see that > r. Since r was an arbitrary positive real number, we see that −1 is a positive infinite element. And of course, −−1 is a negative infinitep element. But now we can continue playing, considering numbers like 2 · −3 and so on..
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